Abstract
We study a scheduling environment that finds many real-world manufacturing applications, in which there is a close connection between a hybrid multiprocessor open shop and multiple parallel identical flow shops. In this environment, there is an extended two-stage open shop, where in one stage we have a set of parallel identical machines, while in the other we have a two-machine flow shop. Our objective is to minimize the makespan, that is, the latest completion time of all jobs. We pursue approximation algorithms with provable performance, and we achieve a 2-approximation when the number of parallel identical machines is constant or is part of the input; we also design a 5/3-approximation for the special case where there is only one machine in the multiprocessor stage, which remains weakly NP-hard. Our empirical experiments show that both approximation algorithms perform much better in simulated instances; their average ratios over the proposed lower bound are around 1.5 and 1.2, respectively.
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Notes
By “consecutively,” we mean the machine processes the jobs one immediately after another without any idling period in between.
Note that the length \(\Delta \) of idle time for machine B guarantees that the new time gap between the start times of \(J_\ell \) on machine A and on machine B is exactly the same as the time gap in the initial schedule \(\pi ^0\) (and thus ensures job processing consecutiveness). Likewise, the length \(\Delta _2\) of idle time for machine C guarantees that the new time gap between the start times of \(J_\ell \) on machine B and on machine C is exactly the same as the time gap in the initial schedule \(\pi ^0\) (and thus ensures job processing consecutiveness).
Compared to the schedule \(\pi ^2\), the revision is on the processing order of the jobs of \(\mathcal{J}^1\) on the three machines.
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Acknowledgements
The authors would like to thank the anonymous reviewers for their many suggestions and comments that helped improve the paper presentation. JD was supported by NNSF China Grants 11971435 and 11501512 and PNSF Zhejiang China Grant LY18A010029. JC and GL were supported by NSERC Canada.
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Dong, J., Chang, J., Su, B. et al. Two-stage open-shop scheduling with a two-machine flow shop as a stage: approximation algorithms and empirical experiments. J Sched 23, 595–608 (2020). https://doi.org/10.1007/s10951-019-00633-7
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DOI: https://doi.org/10.1007/s10951-019-00633-7